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January 7, 1998 \hfill LBNL-41139 \\
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{\large \bf Quantum Nonlocality}
\footnote{This work was supported by the Director, Office of Energy
Research, Office of High Energy and Nuclear Physics, Division of High
Energy Physics of the U.S. Department of Energy under Contract
DE-AC03-76SF00098.}
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Henry P. Stapp\\
{\em Lawrence Berkeley National Laboratory\\
University of California\\
Berkeley, California 94720}
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\begin{abstract}
David Mermin has proposed a criterion for when reasoning about
counterfactuals is acceptable within a quantum context. He suggests
that one step in my recent proof pertaining to quantum nonlocality
fails to meet this criterion, and that the proof therefore has a gap,
insofar as concordance with this criterion is required. Mermin's
criterion was formed by elevating a certain characterization that I had
given of my reasoning. I explain here why the step in question does
conform to my characterization. I also draw a distinction between the
referents of the meaning of a statement and the referents of the proof
of its validity within a particular context. A blurring of this
distinction created the ambiguity mentioned by Mermin.
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{\bf Disclaimer}
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David Mermin$^1$ has nicely summarized and explained the main ideas of my
recent proof$^2$ of the incompatibility of some predictions of quantum
theory with a certain formulation of the notion that no causal
influence of any kind acts backward in time in any Lorentz frame.
Mermin stressed, as I have many times, that this sort of argument involves
reasoning about possibilities for the outcomes of measurements that are not
actually performed, and observed that ``One of the great lessons of
quantum theory is that utmost caution must be exercized in reasoning about
the hypothetical outcomes of unperformed experiments.'' But he agrees with
me that a wholesale rejection of all such reasoning would be crippling to
theoretical analysis, and thus seeks a criterion that would identify
acceptable reasoning within a quantum context. He proposes such a criterion,
and argues that one step of my proof does not fulfill this criterion. Thus
he claims that, within the restrictive framework defined by imposing this
criterion, this step is unjustified, and hence constitutes a ``gap'' in my
proof. Using language reminiscent of Bohr's reply to Einstein, Podolsky, and
Rosen (EPR) Mermin claims, that my reasoning contains ``an essential
ambiguity as regards the meaning of the expression ``statement referring
only to phenomena confined to an earlier time.''
I shall here examine Mermim's arguments and will, I believe, show
why they do not undermine my proof or its conclusion. I divide my
comments into a number of points.
1. Mermin based his criterion for acceptable counterfactual reasoning
on some statements that I had made with different intent.
I said: ``...theoretical assumptions (these will be LOC2, etc.) often
allow us to say with certainty, on the basis of the outcome of a certain
experiment (this will be the + outcome of R2) what `would have
happened' (namely --) if an alternative possible apparatus (e.g., R1) had been
used." And ``Theory (e.g., theoretical assumptions LOC2, etc.) often allow
us to deduce from the outcome (e.g., +) of certain measurements (e.g., R2)
on a system what the outcome of some alternative possible measurements
(e.g., R1) would have been (e.g., --)."
I did not formulate these statements as a ``characterization of when one
can make a meaningful assertion about the result of an unperformed
experiment", but rather as characterizations of my own applications,
including, in particular, the deduction of the line 6 that Mermin objects
to.
But since that criterion does characterize my reasoning at this point,
why does Mermin claim there is a problem?
The answer appears to lie in a certain conflating by Mermin of the ideas of
``meaning'' and ``validity''.
2. Consider two real numbers, $x$ and $y$. The {\it meaning} of ``$x$ is
positive'' does not refer to $y$. But suppose the {\it validity} of this
statement ``$x$ is positive'' is proved within a context defined by the two
assumptions ``$y$ is positive'' and ``$x=y$''. In this case, even though the
basic meaning of ``$x$ is positive'' does not refer to $y$, the {\it proof of
the validity} of this statement, within the particular context defined by
these two assumptions, does refer to $y$.
This example shows that ``meaning'' and ``validity'' can have different
referents.
Mermin's argument that something is wrong with the derivation of line 6,
appears to mix up the ideas of ``meaning'' and ``validity''. With $(S_2)$
defined to be the clause
$(S_2)$ := (if R1 had been performed earlier, instead, it would have given --),
\noindent Mermin writes ``the meaning and validity of $(S_2)$ do not derive
directly from an actual performance of R1 giving --. As a counterfactual
$(S_2)$ only has meaning, according to our criterion, as an inference from
actual results of actual experiments in combination with theoretical
principles''.
Mermin's idea seems to be that $(S_2)$ is meaningless because the reasoning
that leads to it fails to accord with the stated criterion.
But, in the first place, under condition L2, the validity of $(S_2)$ does
follow from the result (+) of the actual experiment R2, plus the
theoretical principles: that is how we got to line 5. Thus the reasoning
leading to line 5 is, it seems to me, in accord with the ``criterion''
for valid reasoning, insofar as this criterion is taken over from my
characterization of my use of counterfactuals.
But what happens under condition L1?
Let statement S be defined by S := (R2+ implies $(S_2)$). Then
LOC2 is the assertion that if, under condition L2, S is true, then S would
be true also if the later free choice had been L1.
So given the principle LOC2 there is no problem obtaining line 6: since
the statement S is true under condition L2, the truth of line 6 follows
directly from LOC2.
A pertinent question, however, is whether LOC2 is actually an expression
only of the idea that no influence of any kind can act backward in time.
This is where the question of ``meaning'' versus ``validity'' enters.
It is true that according to the ideas of logical positivism the meaning
of some statements are defined by how they are verified, and the validation
of S under condition L2 does refer to L2.
But I claim that the present case is like the ``$x$ is positive''
example given above: S has a clear {\it meaning} both in a physical sense and
in a theoretical sense, and this meaning refers only to actualities
and possibilities pertaining to the region R, even though the {\it proof
that S is valid} under certain condition related to events in region L
will of course refer to those events in region L.
Theoretically the meaning of S is specified within a set of conceivably
possible worlds, and S is a constraint within this set of possible worlds:
it imposes exactly the conditions that it describes on the events and
possibilities in R.
Physically the meaning of S is that in a situation where the choices of the
experimenters are effectively free (since these choices could be quantum
free choices) there is a condition or constraint in Nature that does
determine that if the choice had been R2 and the result (+), then if the
free choice had been R1, instead, the outcome would have been (--).
[Note that this conclusion about what ``would have happened'' has been derived
from assumptions that do not include determinism, or any notion that the
outcomes (or probabilities) are determined by hidden variables.]
Given the assumptions LOC1 and the predictions of quantum theory as
physical conditions we have shown that Nature must have this constraint
property S, under condition L2.
The question, then, is whether this condition within Nature pertaining to
actual and possible events in R that does hold under condition L2 would
still be true if the free choice at the later time had gone the other way.
LOC2 asserts, in this context, that this constraint within Nature pertaining
to events and possibilities in R would not be disturbed if the later free
choice had been L1.
LOC2 is therefore, I believe, a bona fide locality condition: it says that
certain specified constraints on Nature's choices pertaining to events and
possibilities in R do not depend on what an experimenter in L freely chooses
to do at some later time.
{\bf Comments}
The notion that the dynamical laws are local and causal in every Lorentz frame
is so ingrained in the thinking of physicists that it may, to many, seem
preferable to abandon or severly curtail logic rather than give it up.
But that attitude is surely unscientific. I shall try here to loosen up
prejudices on this point.
1. It is generally recognized that the Heisenberg picture is more rigorous
than the Schroedinger picture. But in the Heisenberg picture the two different
aspects of the dynamical laws are very distinct: the Heisenberg (commutator)
equations of motion for the local operators are universal, deterministic,
local, and are compatible forward-lightcone causality. But the abrupt change
in the state vector that accompanies the acquisition of new knowledge is none
of these: it depends on the particular state that the universe is in, it is
stochastic, it acts instantly over all of spacetime, and (hence) causal
effects are not confined to forward lightcones.
2. This ``reduction'' of the wave packet (state vector) mentioned above
is an essential part of the mathematical formalism that is used to extract
practical predictions from the quantum principles. The strange properties
of this reduction part was a principal reason why the founders of quantum
theory insisted that the mathematical formalism was to be interpreted as being
about ``our knowledge'', rather than directly about the more extensive reality
of which our knowledge is but a small fragment.
3. The nonlocality theorem being discussed here is part of an effort to
deduce properties of the larger reality from known properties of the small
fragment.
4. In view of the fact that the mathematical formalism that describes
the known properties of the small segment does {\it not} enjoy the properties
of determinism, locality, and forward-causality in all Lorentz frames
that are enjoyed by its (relativistic) classical approximation, it is an
open question whether the full dynamics in the quantum world will be able
to conform to any of these properties of the classical approximation.
Because these nonclassical behaviours play such a key role of the pragmatic
quantum formalism, the a priori presumption would be that the full quantum
dynamics will {\it not} be able to conform to the various classical
idealizations.
5. This certainly does not mean that one should not pursue with utmost vigor
efforts to find logical flaws in theorems that claim to show that certain of
these properties of the classical approximation cannot be maintained in the
general quantum case. But certainly there is no justification of the extreme
position that the rules of logic that philosophers have designed to treat
counterfactual reasoning must, at all cost, be curtailed in such away
as to kill any proof that nature fails to conform to certain properties
possessed the relativistic classical idealizations.
\noindent{\bf References}
1. N.D. Mermin, ``Nonlocal character of quantum theory?'',
American Journal of Physics, preceding article.
2. H.P. Stapp, ``Nonlocal character of quantum theory'',
American Journal of Physics, {\bf 65}, 300-304 (1997).
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